The logarithmic function

Determine the domain of definition of the logarithmic function $f(x) = \log_3(2x - 8)$.

Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x - 5)$. Express the result using interval notation.

Determine the domain of definition of the function $f(x) = \log_{10} \left( \frac{x+3}{|x-2|} \right)$. Pay special attention to the points that make the numerator, the denominator and the argument of the logarithm equal to zero.

Using the laws of logarithms, expand the following expression to write it as the sum and difference of simpler logarithms:

$\log_5 \left( \frac{25x^3}{y} \right)$

Find the values of $x$ and $y$ that satisfy the following system of non-linear equations:

$\begin{cases} x - y = 9 \\ \log_3(x) + \log_3(y) = 3 \end{cases}$

Find the axis intercepts (x-axis and y-axis, if they exist) of the function $g(x) = \ln(x - 2)$.

Given the function $h(x) = \log_{0,5}(x)$, state with reasoning whether it is an increasing or decreasing function across its entire domain, and calculate the exact value of $h(1)$ and $h(4)$.

Describe precisely what geometric transformations (translations) must be applied to the graph of the basic function $f(x) = \ln(x)$ to obtain the graph of the function $g(x) = \ln(x+2) - 3$.

Write the algebraic expression for the inverse function $f^{-1}(x)$ of the exponential function $f(x) = 10^{x+2} - 5$.

From the following graphical representation of the function $f(x) = \log_2(x+2) - 1$, determine analytically its domain, its range, the equation of its vertical asymptote and the exact coordinates of its x- and y-intercepts.

Analyse the symmetry relationship between the functions $f(x) = e^{x-1}$ and $g(x) = \ln(x) + 1$. Observe the attached graph showing both curves and the line y = x (the bisector of the first quadrant). Prove analytically that they are inverse functions by calculating the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$.

Solve the following logarithmic equation by applying the definition of a logarithm:

$\log_2(3x + 1) = 4$

Solve the following logarithmic equation, checking the validity of the solutions obtained:

$\log(x + 2) + \log(x - 1) = 1$

Solve the following logarithmic equation. Make sure to check possible solutions in the original equation to rule out extraneous roots:

$2\log_2(x) - \log_2(x-1) = 3$

In chemistry, the pH of a solution is defined by the expression $pH = -\log[H^+]$, where $[H^+]$ represents the hydrogen ion concentration in moles per litre. Calculate the pH of an orange juice whose hydrogen ion concentration is $[H^+] = 10^{-3,2}$.

The sound intensity level $\beta$, measured in decibels (dB), is calculated using the acoustic formula $\beta = 10 \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ is the standard threshold of hearing. If the take-off of a jet aircraft generates a noise level of $140 \text{ dB}$, what is the intensity $I$ of that sound in $\text{W/m}^2$?

On the Richter magnitude scale, the magnitude $M$ of an earthquake is related to the energy released $E$ (in ergs) by the empirical equation $\log(E) = 11,8 + 1,5M$. Calculate how many times greater the energy released by an earthquake of magnitude $7,0$ is compared to the energy released by one of magnitude $5,0$.